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It's well known that the holomorphic vector fields on a complex manifold form a Lie algebra. In simplest situations, this Lie algebra can be described explicitly.

For example, take $X=\mathbb{P}^n$, then

$H^0(\mathbb{P}^n,T_{\mathbb{P}^n})=\langle z_0\frac{\partial}{\partial z_0},\cdot\cdot\cdot,z_n\frac{\partial}{\partial z_n}\rangle/\langle z_0\frac{\partial}{\partial z_0}+\cdot\cdot\cdot z_n\frac{\partial}{\partial z_n}\rangle$

from which we deduce $H^0(\mathbb{P}^n,T_{\mathbb{P}^n})\cong\mathfrak{sl}_{n+1}$.

In general the description of the Lie algebra structure on $H^0(X,T_X)$ is not easy. I'm particularly interested in the case when $X$ is the cotangent bundle of a flag variety $G/P$, for example, when $X=T^\ast\mathbb{P}^n$. Is there any references in this direction?

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  • $\begingroup$ When $X$ is a flag manifold $G/P$, Baston-Eastwood (1989, p. 49) show that $H^0(X, T_X)=\mathfrak g$ by Borel-Weil-Bott, generalizing your example. Don't know about $X = T^*(G/P)$... $\endgroup$
    – Francois Ziegler
    Commented Sep 1, 2015 at 20:13
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    $\begingroup$ The cotangent bundle of $\mathbb{P}^1$ is a minimal desingularization of the affine quadric surface cone (with a single ordinary double point), e.g., the quotient of the affine plane by the $-1$-action. The Lie algebra is infinite-dimensional. In what terms do you want a description of this infinite-dimensional Lie algebra? $\endgroup$
    – Jason Starr
    Commented Sep 1, 2015 at 21:29
  • $\begingroup$ Just to point out: even considered as a module over the infinite-dimensional algebra of holomorphic functions on the cotangent bundle, still the Lie algebra is not generated by $\mathfrak{g}$. In the case of the cotangent bundle of $\mathbb{P}^1$, the cokernel is one-dimensional, generated by the vector field of "scaling" on the quadric cone. $\endgroup$
    – Jason Starr
    Commented Sep 1, 2015 at 22:16
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    $\begingroup$ @YHBKJ We know (by Mostov 1955 ): $T^*({G/P})\cong G^\mathbb C/P^\mathbb C$ is hyper-Kahler, so it is Calabi-Yau manifold, then its Lie algebra of holomorphic vector fields is abelian hence reductive, see LEMMA 3.3. of jstage.jst.go.jp/article/jmath1948/29/1/29_1_135/_pdf/-char/en $\endgroup$
    – user21574
    Commented Nov 29, 2017 at 6:32
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    $\begingroup$ About my previous comment Biquard- Gauduchon paper numdam.org/article/TSG_1997-1998__16__127_0.pdf is useful. See Calabi paper also if you can read French text eudml.org/doc/82036 or arxiv.org/pdf/math/0011256.pdf $\endgroup$
    – user21574
    Commented Nov 29, 2017 at 6:46

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